It has to be a battery connection issue, as far as I know there is no mechanical linkage. Have a look at all the electrical connections and clean them up with isopropyl alcohol and try again using a new set of batteries.

Speed Grip.
On the speed grip on my SQa there is a small flat button on the base of the grip that locks the shutter finger from pressing on the release button on the front of the camera I have more than once left it in the lock position without realizing it and the grip would advance but not fire the shutter .